Hummel, J. E., & Stankiewicz, B. J. (1998). Two roles for attention in shape perception: A structural descriptionmodel of visual scrutiny. Visual Cognition, 5, 49-79.

Please note: I am not confident this document is the absolutely final (i.e., in print) version of the paper. But it is atleast close.

Two Roles for Attention in Shape Perception:A Structural Description Model of Visual Scrutiny

John E. Hummel Brian J. StankiewiczUniversity of California, Los Angeles

Address correspondence to:John E. HummelDepartment of PsychologyUniversity of California, Los Angeles405 Hilgard Ave.Los Angeles, CA 90095-1563USA

Abstract

This paper presents MetriCat, a model of the human capacity to recognize objects both as membersof a general class (e.g., "chair") and as specific instances ("my office chair"), and of the role ofvisual attention in this capacity. MetriCat represents the attributes of an object's parts and theirrelations in a nonlinear fashion that provides a natural basis for recognition at both the class andinstance levels (Stankiewicz & Hummel, 1996). Like previous structural description models (e.g.,Hummel & Biederman, 1992), MetriCat represents part attributes and relations independently,dynamically binding them into structural descriptions. The resulting representation suggests tworoles for visual attention in shape recognition: attention for binding and attention for signal-to-noisecontrol. MetriCat implements both functions as special cases of a single mechanism for controllingthe synchrony relations among units representing separate object parts. The model accounts for thetime course of class- and instance-level classification, and makes several predictions about therelationships between attention, time, and levels of classification.

Two Roles for Attention in Shape Perception:A Structural Description Model of Visual Scrutiny

Human object recognition is remarkably flexible. Upon viewing a novel instance of aknown object class, it is usually possible to recognize the object immediately as a member of thatclass. For example, in a furniture store we effortlessly recognize the chairs as chairs, the tables astables, and so forth, even if we have never seen those specific chairs or tables before. At the sametime, it remains possible to tell the difference between one chair and another. Although theseobservations conform closely to common sense, it is challenging to understand how the visualrepresentation of shape makes this kind of multi-level classification possible (Marr, 1980).Computational models of object recognition are divided with respect to their capacity to account forclassification at different levels of abstraction. Models based on categorical structural descriptions(Biederman, 1987; Dickenson, Pentland & Rosenfeld, 1992; Hummel & Biederman, 1992;Hummel & Stankiewicz, 1996a) provide a natural account of our ability to recognize objects asmembers of general classes (e.g., “chairs”), but do not provide a clear basis for distinguishingsimilar members of the same class. View-based models (Bülthoff & Edelman, 1992; Bülthoff,Edelman & Tarr, 1995; Edelman, Cutzu & Duvdevani-Bar, 1996; Poggio & Edelman, 1990; Tarr,1995) are adept at classifying objects as individual instances, but are not well-suited to account forour ability to recognize novel instances of known object classes. A clear computational account ofhow both instance- and class-level recognition obtain simultaneously has yet to be developed.

A related question concerns the role of visual attention in shape perception and objectrecognition. A growing body of evidence suggests that we can recognize objects without attendingto them (Tipper, 1985; Tipper & Driver, 1988; Treisman & DeShepper, 1996). However, this doesnot imply that attention plays no role in object recognition. Attention is known to play an importantrole in many basic visual tasks, including feature selection (see Bundeson, this issue) and visualbinding (e.g., Treisman, 1982; Treisman & Gelade, 1980; but see Kanwisher & Wojculik, thisvolume). Using a priming paradigm, Stankiewicz, Hummel and Cooper (1997) recently showedthat attention also plays a role in the visual representation of object shape. Specifically, we foundthat attended images visually prime both themselves and their left-right reflections, whereas ignoredimages prime only themselves (see also Stankiewicz & Hummel, 1997). Thus, although it does notdetermine whether an object will be recognized, attention seems to affect the qualitative form of thevisual representations mediating recognition. This fact suggests the beginnings of an answer to thequestion of how we recognize objects at multiple levels of abstraction: Perhaps attention serves to"tune" the representation of object shape, alternating between coarse, categorical representations(e.g., for class-level recognition) and more precise representations (e.g., for instance-levelclassification). This idea is intuitive, and similar ideas have been proposed before (e.g., Biederman,1987; Hummel & Stankiewicz, 1996a; Marr, 1980; Olshausen, Anderson & Van Essen, 1993). Butto turn this intuitive idea into a falsifiable theory, it is necessary to specify both the form of thevisual representations that are subjected to this attention-based "tuning", and the nature of theattentional mechanisms that perform the tuning.

This paper presents MetriCat (for metric and categorical properties), a model of shaperecognition addressed to the question of how we recognize objects at multiple levels of abstraction(Stankiewicz & Hummel, 1996), and to the role of visual attention in this capacity. MetriCat is anextension of our previous work on structural descriptions of object shape (Hummel & Biederman's,1992, JIM, and Hummel & Stankiewicz's, 1996a, JIM.2). Like JIM and JIM.2, MetriCat representsobject shape in terms of the qualitative properties of volumetric parts (i.e., geons; Biederman, 1987),and the qualitative relations among them. Also like its predecessors, it represents geon attributesand relations independently of one another. These properties give MetriCat the capacity for class-level recognition enjoyed by models based on categorical structural descriptions. However,MetriCat differs from these models in that its representation of shape attributes and relations is notstrictly categorical. Rather, part attributes and relations are represented in a nonlinear form thatemphasizes differences across categorical boundaries without discarding all metric informationwithin those boundaries. This property permits the model to represent and match object shape at

multiple levels of numerical specificity, and provides a natural basis for classifying objects atmultiple levels of abstraction (Stankiewicz & Hummel, 1996).

This approach to the representation of object shape suggests two separate but related rolesfor attention in shape perception. Coding part attributes independently of their relations makes itnecessary to bind them dynamically (i.e., actively) into part-based groups (Hummel & Biederman,1992). Attention is known to play a central role in binding independent visual attributes (seeSchneider, 1995 for a thorough review), and therefore figures prominently in the generation ofstructural descriptions from object images (Hummel & Biederman, 1991, 1992; Hummel &Stankiewicz, 1996a; Stankiewicz, et al., 1997; see also Logan, 1994). Like its predecessors,MetriCat requires attention to bind attributes into parts-based structural descriptions (although itimplements attention more explicitly than either JIM or JIM.2). It also uses attention to generatethe metrically-detailed representations that permit classification at progressively finer levels ofspecificity (e.g., from chair, to office chair, to my office chair). Both these operations—attributebinding, and the generation of detailed representations of shape—are performed by a singlemechanism: namely, by enhancing the ability of object parts to inhibit one another in thecompetition for binding and processing resources. The model's attention mechanism thus unifiesthe binding and selective processing functions performed by visual attention (see Schneider, 1995).(However, we should note that the model is not intended as a general theory of visual attention.Among other things, it is not explicitly addressed to phenomena such as location-based visualsearch, visual neglect following brain damage, etc. Rather, our current focus is strictly on the role ofattention in the representation and classification of object shape.) The resulting model makestestable predictions about the roles of time and attention in recognition at different levels of visualspecificity, suggests a specific basis for intelligent search for diagnostic features for subordinate-and instance-level classification, and generates testable predictions about the relationship betweenlevels of classification and the effects of viewpoint on recognition performance.

Theories of Object Recognition

Recent theories of object recognition fall into two general classes with respect to their abilityto account for shape classification at different levels of abstraction. One class of theories assumesthat objects are represented as structural descriptions specifying an object's parts in terms of theircategorical attributes and relations to one another (Biederman, 1987; Dickenson, et al., 1992;Hummel & Biederman, 1992; Hummel & Stankiewicz, 1996a). For example, a table might berepresented as a horizontal slab on top of four vertical posts; a coffee mug might be represented asa squat vertical cylinder with a curved cylinder end-attached to its side (see Biederman, 1987).Categorical representations of this type have several desirable properties, including robustness tonoise and variations in viewpoint, and a natural capacity to generalize over variations in object shape(see Biederman, 1987; Pinker, 1984). As a consequence, they provide a natural account of thehuman ability to recognize objects in novel viewpoints (Biederman & Cooper, 1991a, 1992;Biederman & Gerhardstein, 1993; Cooper, Biederman & Hummel, 1992), our ability to recognizeobjects in the presence of noise or occlusion, and our ability to recognize novel instances of knownobject classes (Biederman, 1987; Clowes, 1967). Some structural description models also accountfor a variety of more subtle properties of human shape perception, including the role of convexparts in shape perception (Biederman, 1987; Biederman & Cooper, 1991b; Hoffman & Richards,1985; Tversky & Hemenway, 1985), the role of categorical relations in shape perception and objectrecognition (Hummel & Stankiewicz, 1996b; Saiki & Hummel, 1996), and the role of time andattention in shape perception and object recognition (Hummel & Stankiewicz, 1996a; Stankiewicz etal., 1997).

For the same reason that they support recognition at the level of object classes, categoricalstructural descriptions are insufficient to make some within-class distinctions. For example, acategorical structural description of a robin would be indistinguishable from a categorical structuraldescription of a blue jay: On the basis of such a description alone, both objects would simply berecognized as birds. In such cases, people search for distinguishing features (such as the red breast

on the robin, or the blue crest on the jay) as a basis for classifying these objects at the subordinate-levels of "robin" and "blue jay" (Biederman, 1987; Biederman & Schiffrar, 1987). But while suchfeatures play a critical role in subordinate-level recognition, it is unlikely that they constitute ouronly basis for distinguishing structurally similar objects. Although it is difficult, we candiscriminate objects that differ only in their metric properties (Bülthoff & Edelman, 1992; Edelmanet al., 1996; Hummel & Stankiewicz, 1996b).

In contrast to structural description models, view-based models represent objects as holistic"views" specifying the 2-D coordinates of their features as they appear in specific views (Bülthoff& Edelman, 1992; Bülthoff at al., 1995; Edelman et al., 1996; Poggio & Edelman, 1990; Tarr, 1995;Vetter, Poggio & Bülthoff, 1994). This approach differs from the structural description approachin that (a) there is no explicit decomposition of an object into its parts, and (b) an object's featuresare represented in terms of their numerical coordinates in the view, rather than their categorical (e.g.,"above/below") relations to one another. This kind of representation is metrically precise andinformation-rich, making it suitable for distinguishing structurally similar objects (Bülthoff et al.,1995; Tarr, 1995). There is substantial support for the idea that some instance-level classificationtasks—most notably face recognition—are mediated by this type of holistic representation of objectshape (e.g., Farah, 1992; Tanaka & Farah, 1993). In addition, there is evidence for the role of view-like representations in our capacity to recognize objects without attending to them (Stankiewicz etal., 1997), although the views implicated in automatic recognition may differ substantially from themetrically precise, information-rich representations postulated in current view-based models (seeHummel & Stankiewicz, 1996a).

If the view-based approach provides a general account of instance-level classification (i.e.,beyond the case of faces and similar objects), then our capacity for multi-level classification maysimply reflect the simultaneous operation of both views (for instance recognition) and structuraldescriptions (for class recognition). However, not all subordinate- or instance-level recognition isperformed on the basis of holistic views. For example, Biederman and Schiffrar (1987) showedthat categorical features play the central role in chick-sexing1, a decidedly subordinate-levelclassification task, and Hummel and Stankiewicz (1996b) report evidence against the role of holisticviews in the recognition of structurally-similar objects. Thus, although holistic views likely play animportant role in face recognition, and although diagnostic categorical features likely play animportant role in subordinate- and instance-level recognition, neither proposal provides a completeaccount of our ability to recognize objects at multiple levels of abstraction. As such, we believe it isworth considering alternative explanations for this capacity.

A Unified Model of Instance- and Class-Level RecognitionAs noted previously, categorical representations (as postulated in structural description

theories) naturally support recognition at the entry- or class-level, while metrically-richrepresentations (as postulated in view-based theories) naturally support instance- and subordinate-level recognition. MetriCat is based on a single representation that captures both these properties.The basis of MetriCat is a simple extension of the representations postulated in categoricalstructural description models. A categorical representation entails a nonlinear mapping from astimulus domain to the perceptual representation of that domain. For example, consider an imagecontaining two figures, A and B. The relation Above(A, B) is strictly categorical if it evaluates totrue (or 1) for all locations of A and B such that YA (the location of A on the vertical axis, Y) isgreater than YB, and to false (or 0) for all locations such that YA ≤ YB. In this case, the mappingfrom the stimulus domain (coordinates in the image) to the relation is a step function (i.e., Above(A,B) is 1 for all (YA - YB) > 0, and 0 for all (YA - YB) ≤ 0). A step function is strictly categorical inthe sense that its value (0 or 1) changes only at the categorical boundary (i.e., the derivative isinfinite at the categorical boundary and zero everywhere else), discarding all information on eitherside of the boundary. However, a step is a special case of a more general class of nonlinear

1Chick-sexing is the task of deciding, on the basis of the appearance of the genetalia, whether a given chick is maleor female.

functions whose derivative is a maximum over some point (such as a categorical boundary), andreduced elsewhere. A common example is the logistic function:

y =1

1+ e−x . (1)

Like a step function, y changes fastest over a single point in x (specifically, x = 0), but unlike a step,it continues to change (although progressively less) as x deviates further from zero. This "softer"nonlinearity is meaningfully categorical in that, like a step function, it emphasizes differences acrosscategorical boundaries (e.g., from Above to Below). (Indeed, in the literature of categoricalperception, this kind of function is the behavioral manifestation of categorical perception; see, e.g.,Foster & Ferraro, 1989). But the logistic function has the advantage that it does not discard allinformation on either side of the boundary.

We argue that extant structural description models are insufficient for instance-levelrecognition largely because they are based on categorical properties defined by step functions.MetriCat uses logistic (rather than step) nonlinearities to represent categorical properties (such aswhether one part is above or below another, and whether a geon's major axis is straight or curved).In other respects, MetriCat is like its predecessors, JIM and JIM.2: It represents part attributes andrelations explicitly and independently, and binds those properties into relational structuresdynamically; that is, the representations in MetriCat are structured rather than holistic (see Hummel& Biederman, 1992). The resulting representations have all the advantages of a categoricalstructural description, but also permit fine metric discriminations when necessary. In combinationwith an appropriate set of routines for matching these representations to object memory, the result isunified account of both class- and instance-level recognition. Moreover, this approach torepresenting object shape suggests a unifying account of two seemingly different functions ofvisual attention. In MetriCat, a single mechanism controls attention for binding (e.g., as discussedby Treisman and others) and attention for signal-to-noise control (e.g., as discussed by LaBergeand others).

Assumptions and Relation to Prior WorkThe focus of the MetriCat model is the representation of object shape and the operations

that match those representations to memory. (Specifically, it is an extension of the upper layers ofJIM and JIM.2, which are responsible for shape representation and object classification; the lowerlayers of these models are responsible for image segmentation, etc.) MetriCat takes a description ofan object's parts and their spatial relations as input. Rules for segmenting images into part-basedgroups, and for recovering the parts' attributes and spatial relations are described elsewhere(Dickenson et al., 1992; Hummel & Biederman, 1992; Hummel & Stankiewicz, 1996a), so for thepurposes of the current model, we shall simply assume that these operations have taken place.

OverviewAs input, MetriCat takes a numerical representation of a geon's shape attributes and

relations to other geons (Figure 1, "Input"). One such pattern is given for each geon in an object.In the model's first layer (Figure 1, "Layer 1"), separate collections of units respond to: (1) theparallelism of a geon's sides, (2) the curvature of its major axis, (3) the curvature of its cross section,(4) its aspect ratio, the (5) pointedness of its major axis (i.e., is the major axis pointed, as in a cone,or truncated, as in a cylinder or funnel), (6) whether the geon is above or below any other geons,and (7) whether it is beside or centered on any other geons. (This is an admittedly simplified set ofattributes. Among other potentially important properties, we are omitting the geons' connectednessrelations. For a richer set of geon and relation descriptors, see Biederman, 1987.) Shape attributesand relations are represented independently, in the sense that separate collections of units representseparate attributes. Units are bound into geon-based sets by synchrony of firing (Hummel &Biederman, 1992): If two units "fire" (i.e., become active) in synchrony with one another, then theyare treated as properties of the same geon; if they fire out of synchrony, then they are treated asbelonging to separate geons. Synchrony and, more importantly, asynchrony are established by

Axis:Cross sc.:

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Image Part Decomposition

P r e - p r o c e s s i n gHummel & Biederman, 1992Hummel & Stankiewicz, 1996a

Input:Geon Attributes &

Relations

Attribute Coding:Nonlinearity over

Categorical Boundaries

Layer 1:Geon Attributes

& Relations

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Assemblies(Gaussian Receptive Fields)

Layer 3:Objects

(Gaussian Receptive Fields)

... ......

...

...

Grouping by Synchrony(Oscilatory Gates)

Axis Cross sc Above Beside

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Figure 1. The MetriCat architecture. As input, the model takes a numerical description of an object in terms of itsconstituent geons and their interrelations (Input). Geon attributes grouped by synchrony induced by the action ofoscillatory gates (Grouping by Synchrony). Each geon is characterized by five shape attributes, parallelism ofmain axis (Ax.Paral.), curvature of main axis (Ax.Cur.), curvature of cross section (XSn.Cur.), aspect ratio(Aspect), and pointedness of main axis (Ax.Point), and two spatial relations, above/below (Above) andcentered/beside (Beside). The numerical value of each attribute is passed through a logistic nonlinearity (AttributeCode), the result of which is represented on a bank of 50 units in the model's first layer (Layer 1). Patterns ofactivation in Layer 1 activate Geon Feature Assembly (GFA) units with Gaussian receptive fields in Layer 2.Patterns of activation on GFA units activate object units with Gaussian receptive fields (Layer 3). GFA and objectunit have Gaussian receptive fields with different standard deviations.

means of oscillatory gates associated with the geons: The gates interact so as to "open," sending theproperties of the corresponding geon to the model's first layer, one at a time. There is evidence forsynchrony-based binding in biological visual systems (see König & Engel, 1995, for a thoroughreview).

In broad strokes, the model works as follows. When an object is presented for recognition,it's geons begin to fire out of synchrony, generating separate patterns of activation on the attributeand relation units in Layer 1. In the best case, one geon will fire at a time, but as described shortly,the geons' ability to fire cleanly out of synchrony is a function of how much "attention" is directedto the object. The model's second layer (Layer 2) consists of Geon Feature Assembly units (GFAunits; Hummel & Biederman, 1992) that take their inputs from the attribute and relation units,thereby responding to particular geons in particular relations to other geons. The model's thirdlayer (Layer 3) consists of object units that take their inputs from the GFA units. Object units sumtheir inputs over time, "reassembling" a collection of GFAs (geons in particular relations) into arepresentation of a whole object.

GFA and object units (classifier units) have Gaussian receptive fields (RFs) in their inputspaces, where the mean of the Gaussian determines a unit's preferred input pattern, p, and thestandard deviation, σ, determines the unit's tolerance for deviations from that pattern (see Poggio &Girosi, 1990). GFA units have RFs in the space of geon attributes and relations, and object unitshave RFs in the space of GFAs. Classifier units with wide RFs (i.e., large σ) respond to a widerrange of inputs than units with narrow RFs; that is, wide units are better able to tolerate to deviationsfrom their preferred patterns than are narrow units. This variable-tolerance encoding has severalimportant implications. First, classifier units with wide RFs respond to whole classes of objects,whereas units with narrow RFs tend to respond selectively to individual instances (Stankiewicz &Hummel, 1996). Second, wide (class-level) units are able to respond earlier in processing thannarrow (instance-level units) units (as elaborated shortly). As a result, MetriCat predicts thatobjects will be recognized at the class-level before they are recognized as individual instances. Andthird, wide units are more robust to noise in their input vectors, and are therefore better able torespond when an object is not fully attended: MetriCat predicts that greater attention (e.g., visualscrutiny) is required to recognize objects at the instance level. Each of these properties is illustratedand elaborated in the Simulations section.

Representation of Attributes and RelationsEach attribute (or relation) is represented by a bank of units that coarsely code the numerical

value of that attribute (Figure 2). The coded value for any attribute is obtained by passing theattribute's raw numerical value through a logistic function (Hummel & Stankiewicz, 1996b;Stankiewicz & Hummel, 1996). For example, consider the relation Above(i, j), which codes thelocation of geon i relative to geon j on the vertical image axis, Y. PY(i, j) is the scaled location of irelative to j on Y:

PY(i,j) = (Yi - Yj)/(lYi+ lYj), (2)where Yi and Yj are the Y-coordinates of the centroids of i and j, respectively, and lYi and lYj are thelengths of i and j along Y. PY(i, j) is unbounded. It will be negative whenever i is below j, andpositive whenever i is above j. Scaling P by the parts' lengths makes it scale-invariant: P will notchange with the absolute size of the object's image. Human object recognition is also scale-invariant in this way (Biederman & Cooper, 1992). For an image of a fixed size, PY(i, j) changeslinearly with the location of geon i relative to geon j along Y. Above(i, j), the above/below locationof i relative to j that is used for shape classification, is computed by passing PY(i, j) through thelogistic function:

Above(i, j) =

1

1+ e − PY(i, j ), (3)

where is a scaling constant. Although PY(i, j) is unbounded, Above(i, j) is bounded between 0and 1; it will be less than 0.5 whenever i is below j (i.e., when P < 0) and greater than 0.5 whenever iis above j(when P > 0). Like a categorical relation, Above() changes fastest about the categorical

boundary, P = 0 (where Yi = Yj). However, as noted previously, Above() does not discard all metricdifferences within categorical boundaries. determines the steepness of Above() about thecategorical boundary P = 0; when = ∞, Above() is a step function that evaluates to 0 for all P < 0and to 1 for all P > 0.

0.0

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erre

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Figure 2. Detail of the representation of object attributes (in Layer 1), illustrated with the relation Above. Theraw value of an attribute or relation is passed through a logistic nonlinearity, and the resulting value is coarselycoded on 50 units with Gaussian receptive fields in the space of logistic values (0..1). Units are depicted ascircles, receptive fields are depicted as Gaussian curves attached to the circles, and activation as shades of gray,with more active units depicted in darker shades. The relation represented corresponds to PY(i, j) = 0.5, or "geon i

is above geon j at a distance that is 1/2 the sum of their heights."

The left-right relation, Beside, is computed in the same way as the Above relation, except thatleft/right distinctions are discarded, so the function effectively computes centeredness vs. off-centeredness with respect to the horizontal axis:

Beside(i, j) = 2

1

1+ e− P X ( i , j )− 0.5

. (4)

As a result, Beside(i, j) is invariant with left-right reflection (Hummel & Biederman, 1992). Itevaluates to zero when geon i is exactly centered on geon j (on the X axis), and to values closer to1.0 as i moves farther to the left or right j.

Equations (3) and (4) express the functions for computing all attributes in MetriCat. Eq.(3) expresses the form for any attribute with two continuous regions separated by a categoricalboundary. These include the Above relation (one geon can be more or less extreme in its locationeither above or below another geon) and aspect ratio (a geon can be more or less squat [e.g., a coinis a squat cylinder] or more or less elongated [e.g., a pipe is an elongated cylinder]). Eq. (4)expresses the form for an attribute with one continuous region and one singularity. For example, inthe case of the Beside relation, off-center (or "beside") is a continuous region and centered is asingularity (a geon can be more or less distant from the midline of another, but there is only one

point where it is exactly centered). Most of the properties in MetriCat are of this latter variety.These include parallel sides (singularity) vs. non-parallel sides (continuous), straight cross section(singularity) vs. curved cross section (continuous), pointed major axis (singularity) vs. truncatedmajor axis (continuous), and straight major axis (singularity) vs. curved major axis (continuous).

The model's input layer consists of seven banks of units (one bank for each attribute), eachwith 50 units. Units within a bank respond to logistic values given by (3) and (4) (depending onthe attribute). Units have Gaussian receptive fields over the range of logistic values (0..1) withvarying centers (means) and receptive field sizes (standard deviations). As illustrated in Figure 2,the result is a population coding for the logistic value of each attribute (Stankiewicz & Hummel,1996).

Model Operation

The model's operation is described here only in general terms. The details of its operation(including equations and operating parameters) are given in the Appendix.

Oscillators and Visual Attention: An object's geons are induced to fire out ofsynchrony by means of a collection of oscillatory gates. There is one gate associated with eachgeon. By virtue of the interation between an excitatory unit (Ei) and an inhibitory unit (Ii), eachgate, i, oscillates between a state of activity (Ei > 0), in which the gate is "open", and inacitvity (Ei =0), in which the gate is "closed" (see Hummel & Stankiewicz, 1996a; von der Malsburg & Buhman,1992). When a gate is open (i.e., Ei, > 0), the attributes of the corresponding geon are representedon the attribute units. If more than one gate opens at a time, then the attributes of multiple geonswill be superimposed on the attribute vectors. The result is a binding error (or "superpositioncatastrophe"; von der Malsburg, 1981), because it is impossible to determine which attributesbelong together as properties of the same geon. The gates act to prevent such errors by inhibitingone another, thereby opening (or "firing") at different times (i.e., out of synchrony with oneanother; Figure 3c).

Time

a) α = 0.0

E4

E1

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E4

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E4

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Figure 3. Simulation results illustrating the effect of attention (α) on MetriCat's ability to make separate geons

fire out of synchrony. Each frame shows the activation of four gates (E1..E4) as a function of time and α.

Activation (Ei = 0..1) is depicted on the ordinate of each curve. (a) With α = 0 (no attention) all four gates (geons)

fire in synchrony with one another. (b) With α = 0.5 (limited attention), the gates fire out of synchrony, but there

is substantial overlap in their activity. (c) With α = 1.5 (strong attention), the gates fire cleanly out ofsynchrony.

The model's ability to keep separate geons firing out of synchrony with one another isdependentr on the gates' ability to inhibit one another. In turn, the lateral inhibition between thedifferent gates is modulated by a global parameter, α. Figure 3 illustrates the effect of α on thegates. When α is large (e.g., 1.5), the gates inhibit one another strongly and therefore fire cleanlyout of synchrony (Figure 3c); when α is zero, the gates cannot inhibit one another at all, andtherefore never fire out of synchrony (Figure 3a); at intermediate values of α, the gates firesomewhat out of synchrony (Figure 3b). We assume that α is proportional to the amount ofattention directed to an object. Our claim is that the function of attention (at least in the context of

object recognition) is to enable separate groups of units to fire out of synchrony with one another.In so doing, attention serves both to control the dynamic binding of geon attributes and relations(Hummel & Biederman, 1991, 1992; Hummel & Stankiewicz, 1996a), and to control the signal-to-noise ratio of attended objects (e.g., LaBerge & Brown, 1989): Note that the pattern of asynchronyis cleaner for "strongly-attended" objects (Figure 3c) than for “weakly-attended” objects (Figure3b). The effect of α on model performance is discussed in greater detail in the Simulations section.

Coarse Coding of Attribute Values: Each geon attribute is represented as a pattern ofactivation distributed over a bank of 50 attribute units (Figure 2). Attribute units have overlappingreceptive fields over the range of logistic values of the corresponding attribute (Eq. (3) or (4),depending on the attribute). A complete geon is represented by a 350-dimensional (50 units X 7attributes) vector, a.

Part Classification: GFA units have Gaussian RFs in the 350-dimensional space ofattribute units. Each GFA unit receptive field has a mean, p, a vector that corresponds to the unit'spreferred stimulus pattern, and a standard deviation, σ, which defines the width of the unit's receptivefield (i.e, its tolerance for deviations from p). The simulations reported here were run with threesizes of GFA unit RFs, = 0.10, 0.25, and 0.35. Units with small respond to very specificattribute vectors (i.e., to parts with specific shapes in specific relations); those with larger respondto a broader range of vectors (and thus to a broader range of part shapes and relations).

Object Classification: GFA units sum their outputs over time. Object units haveGaussian RFs in the space of GFA unit outputs. As a result, object units respond to collections ofGRA units, reassembling an object's parts into a representation of the object as a whole (seeHummel & Biederman, 1992). There were two sizes of object unit RFs in the simulations reportedhere, = 0.10 and 0.5.

Simulations

Simulation ProcedureWe trained the model to recognize 16 objects, comprising four instances of each of four

classes. As illustrated in Figure 4, we will refer to the classes as teapots, lamps, cars, and nonsenseobjects. Table 1 shows the attribute values of each part of each object. Each instance differed fromthe next nearest member of its class by 0.5 raw2 units on one critical attribute dimension (Table 1).The teapots differed in the aspect ratio of the truncated cone (raw values 1.0, 1.5, 2.0 and 2.5 forteapot1..teapot4, respectively), the lamps differed in the degree of non-parallelism of the shade (rawvalues 1.0..2.5), the cars differed in the aspect ratio of the chassis (-2.5..-1.0), and the nonsenseobjects differed in the curvature of the curved cylinder (1.0..2.5).

Training took place in two steps. The model was first trained to classify the individualobject parts by presenting each part one at a time, and running the model for 100 iterations. At theend of this time, we compared the attribute unit vector, a, to the preferred vectors, pi, of all existingGFA units, i. If the Euclidean distance between a and pi was greater than 0.05 for all i, then wecreated three new GFA units (one at each σ) with p = a. Next, we trained the object units bypresenting each object part for 100 iterations and allowing it to activate all the GFA units. After allthe parts of an object had been presented, we created two object units (one at each σ) whose means(p) were set to the value of the GFA output vector. All the simulations reported in the next sectionwere run by presenting a single object as input and allowing the model to run for 500 iterations.Except where noted otherwise, α was 1.5 in all simulations. The data reported are means over tensimulation runs. Simulations were run with multiple objects, but because the results were alwaysqualitatively very similar for all objects, each set of results is reported in terms of a set of runs withone object.

2"Raw" units are units that serve as the input to (rather than the output of) the logistic functions (Eq. (3) and (4)).

TeapotDistinguishing property:

Aspect ratio of truncated cone

CarDistinguishing property:Aspect Ratio of Chasis

LampDistinguishing property:Degree of non-parallelism

in Shade

Nonsense objectDistinguishing property:

Axis curature of curved cylinder

Figure 4. Graphic depiction of the four object classes MetriCat was trained to recognize.

Multi-level ClassificationThe most basic test of the model's capacity for multi-level classification is its response to a

stimulus on which it was trained. If MetriCat can recognize objects at both the instance and class-levels, then the following results should obtain. Given a stimulus such as car1 (the first member ofthe category car), the model should activate both the instance-level (i.e., small σ) and class-level (i.e.,large σ) object units recruited during training with that stimulus. It should also activate the class-level units recruited for different instances in the same class (here, car2, car3, and car4), but it shouldnot activate the instance-level units recruited for those objects. It should activate neither the class-nor instance-level units for any other objects (i.e., teapot1..nonsense-object4). Figure 5 shows themodel's response to the stimulus car1. Figure 5a shows the mean responses (over ten runs) of theinstance-level unit (triangles) and class-level unit (circles) recruited for car1. Figure 5b shows themean responses of the instance- and class-level units for car2, the nearest other instance of the sameobject class. (Recall that car1 and car2 differ by only 0.5 raw units in the aspect ratio of one part.)Figure 5c shows the mean responses of the instance-level and class-level units for the central-mostinstance in the class lamp (lamp1). These results indicate that the model correctly recognized car1at both the instance- and class-levels of abstraction: Both the instance- and class-level unitsrecruited for car1 became active; the class-level unit, but not the instance-level unit for car2 becameactive; and none of the units corresponding to lamp1 became active.

Figure 5 also shows the time course of recognition at the class- and instance-levels. Notethat class-level units (circles) become active faster than instance-level units (triangles): Consistentwith human recognition performance (Jolicoeur, Gluck & Kosslyn, 1984; Rosch, Mervis, Gray,Johnson, & Boyes-Braem, 1976), MetriCat recognizes objects at the class-level faster than itrecognizes them at the instance-level. This property is a natural consequence of MetriCat's multi-level approach to object recognition. Recall that class-level units have wider receptive fields (largerσ) than instance-level units. (This is true of both object units and GFA units.) As a result, a givendeviation (distance) between a stimulus pattern (a, in the case of GFA units and o in the case ofobject units) and a unit's preferred stimulus pattern (p) has a greater impact on the response of a

Table 1. Attribute values of the 16 objects MetriCat was trained to recognize.a

Object Part AxisParallelism

Cross-Section

AspectRatio

AxisCurvature

AxisTrucation

Above Beside

NonsenseObject

CurvedCylinder

4 -4 4 1 , 1 . 5 ,2 , 2 . 5

0 4 4

Brick 0 4 -4 0 4 -4 4Cone 4 -4 4 0 0 4,-4 0Sphere 4 -4 0 0 0 4 0

Teapot TruncatedCone

4 -4 1 , 1 . 5 ,2 , 2 . 5

0 4 4,-4 4

CurvedCylinder

0 -4 -4 4 4 0 4

Sphere 4 -4 0 0 0 4 0Cylinder 0 -4 4 0 4 0 4

Lamp TruncatedCone

1 , 1 . 5 ,2 , 2 . 5

-4 4 0 0 4 0

VerticalBrick

0 4 4 0 4 4,-4 0

Sphere 4 -4 0 0 0 0 4HorizontalBrick

0 4 -4 0 4 -4 0

Car TruncatedPyramid

4 4 -1 .0 , -1 .5 ,-2 .0 , -2 .5

0 4 4 0

Brick 0 4 4 0 4 4,-4 4HorizontalCylinder

0 -4 -4 0 4 -4 4

Sphere 4 -4 0 0 0 -4 0Sphere 4 -4 0 0 0 -4 4

aThe four instances of each class are distinguished by their values on one critical attribute (depictedin bold in table cells).

narrow (instance-level) unit than on the response of a wide (class-level) unit3. Stimuli tend todeviate more from the trained patterns (i.e., the units' preferred patterns) earlier in processing thanlater in processing. At the level of the GFA unit inputs (a), this early deviation results from the timeit takes for an object's geons to get out of synchrony with one another (see Figure 3b and 3c),causing the properties of different geons to be "mixed" early in processing (as elaborated below).At the level of object unit inputs, this early deviation results both from the initially weak activation ofthe GFA units and from the time it takes for all an object's geons to have the opportunity to fire.Early in processing, the GFA outputs, Oi, will be zero for all GFA units, i, whose preferred geonshave not yet fired, causing o to differ from the object unit's p. Here, too, class-level units, whichhave wide receptive fields, will respond more strongly to these imperfect inputs than instance-levelunits, with their narrow receptive fields.

Recognition of Novel Instances of Known ClassesPeople easily recognize novel instances of known object classes, appreciating both that the

novel instance both belongs to the familiar class and that it nonetheless differs from all previous

3Consider two Gaussian distributions, G1 with σ1, and G2, with σ2, where σ1 > σ2 (i.e., G1 is wider than G2),and let G1 and G2 be normalized to have equal heights at their means. For any distance, d > 0, from their means,the height of the wide Gaussian, G1, will be greater than the height of the narrow Gaussian, G2. That is, G1(d) >G2(d) when d > 0 [G1(d) = G2(d) when d = 0].

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Figure 6. Response to a novel instance of a knownclass. (a) Activation of the class (circles) and instance(triangles) object units recruited in response to thestimulus object as a function of time (iterations duringthe simulation). (b) Activation of the class andinstance object units recruited in response to a differentmember of the same class.

Figure 5. Response of various object units to thestimulus car1. (a) Activation of the class (circles) andinstance (triangles) object units recruited in response tocar1 as a function of time (iterations). (b) Activation ofthe class and instance object units recruited in responseto car2 as a function of time. (c) Activation of the classand instance object units recruited in response to lamp1as a function of time.

instances of that class. We tested MetriCat's ability to do this by presenting it with a new memberof the class nonsense object. We created this instance by distorting nonsense object1. Distortionswere introduced by changing the raw values of each attribute by a random amount between -0.01and 0.01 (20% of the distance between adjacent members of the class on the critical attribute).Figure 6 shows the model's response to the resulting novel instance. Circles indicate the meanactivation of the class-level unit for nonsense-object1, and triangles indicate the response of theinstance-level unit for that object (which is the closest instance to the novel one). Note that themodel correctly recognized the novel instance as a member of the same class, but it did not mistakeit for a familiar instance (as indicated by the low activation of the instance-level unit).

Sensitivity to NoiseMetriCat is subject to intrinsic system-level noise introduced by the operations that establish

asynchronous firing of an object's geons (Figure 3). This noise manifests itself as randomvariations in the patterns of activation generated on the attribute units. MetriCat's ability torecognize trained objects in spite of this noise shows that the model is at least somewhat noise-tolerant. To further observe the model's sensitivity to noise, we tested its ability to recognize stimulicorrupted by variable, random input noise. We generated these stimuli by starting with a trainedobject and varying each attribute value by a random amount between -0.01 and 0.01. The noise wasre-randomized on each iteration. The resulting variable noise can either be thought of as stimulusnoise, such as static in the image, or as additional system-level noise. Because the noise was re-randomized on each iteration, the mean value of each attribute (over several iterations) tends toapproach the value of that attribute in the original stimulus. If MetriCat can tolerate this kind ofnoise, then it should recognize the stimulus at both the class and instance levels (althoughrecognition should take longer than in the no-additional-noise case). Figure 7 shows the model'sresponse to a noisy version of nonsense-object1. As expected, the model recognized the stimulus atboth the class and instance levels, although classification took longer and did not reach as high alevel as in the basic simulations reported above. These results are especially interesting incomparison with the results of the simulations for Novel Instance simulations reported above. Theonly procedural difference between the current simulations and the Novel Instances simulations isthat the current simulations varied the noise on every iteration, whereas the deviations in the NovelInstances simulations did not vary. MetriCat responded appropriately in both cases: It treated theconstant noise as a property of the object, classifying the stimulus as a novel exemplar of a knownclass; and it treated the variable noise as noise, recognizing the stimulus as a familiar (albeit noisy)instance of a familiar class.

Attention in Instance-level ClassificationIn MetriCat, attention modulates the lateral inhibition that causes an object's geons to fire

out of synchrony. If two (or more) geons drift into synchrony with one another, then the propertiesof one will interfere with the interpretation of the other. As noted previously, GFA and object unitsare not all equal in their capacity to tolerate this type of processing noise: Units with wide RFs(large σ) are better-suited to tolerate deviations from their trained patterns than are narrow (small σ)units. At the same time, narrow units are better suited to discriminate objects at the instance-levelthan are wide units. Together, these properties predict that, with reduced attention, class-levelrecognition will be possible but instance-level recognition will not (especially if the object isdepicted in a novel view; see Hummel & Stankiewicz, 1996a). This property is illustrated in Figure8a, which shows the response of the model to nonsense-object1 when the attention parameter, α,was set to 0.5 (rather than 1.5, the value used in the previous simulations). Note that MetriCatrecognized the stimulus as a member of the class nonsense-object, but did not recognize it as theinstance nonsense object1.

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Figure 8. Response to a trained stimulus under reduced attention. Circles depict the activation of the class-levelunit recruited in response stimulus. Triangles depict the response of the instance-level unit recruited in responseto the stimulus. (a) Model response with reduced attention (α = 0.5). (b) Model response without attention (α =0).

Figure 8b shows the model's performance with α set to zero—that is, when the stimulus isnot attended at all. Here, the object is recognized neither at the instance nor class levels. This resultseems to predict that object recognition requires attention. However, recall that MetriCat isintended, not as a stand-alone model, but as a component of the upper layers of Hummel andStankiewicz's (1996a) JIM.2. Specifically, Layer 1 of MetriCat corresponds to the IndependentGeon Array (IGA) of JIM.2, and the classification layers of MetriCat correspond to theclassification layers of JIM.2. Like Layer 1 of MetriCat, the IGA represents shape attributesindependently, is dependent on dynamic binding, and is responsible for recognition by structuraldescription. However, JIM.2 has an additional component, the Substructure Matrix (SSM), thatdoes not require dynamic binding. For familiar object views, the SSM is sufficient for rapidrecognition without attention. The IGA and SSM represent object shape in qualitatively differentways. JIM.2 predicts, not that unattended objects will not be recognized, but that recognition ofunattended objects will differ qualitatively from the recognition of attended objects (see Hummel &Stankiewicz, 1996a). Stankiewicz et al. (1996; see also Stankiewicz & Hummel, 1997) reportexperimental findings that are strikingly consistent with the specific predictions of JIM.2. Forsimplicity, we have left the SSM out of MetriCat, but in principle, the SSM could be part of

MetriCat, just as it is part of JIM.2. Thus, we take the simulation results in Figure 8b to predict thatstructural description (and recognition on the basis of a structural description) will require attention,not that recognition at all will require visual attention.

Discussion

MetriCat is a structural description model of multi-level shape classification. Like itspredecessors, JIM and JIM.2, MetriCat represents object shape as a collection of parts in particularspatial relations. Part attributes and relations are represented independently, and are dynamicallybound into structural descriptions by synchrony of firing. However, MetriCat departs from itspredecessors in that its representation of part attributes and relations is not strictly categorical.MetriCat's representations are categorical in the sense that they are nonlinear over categoricalboundaries in attribute dimensions, but they are metric in that they preserve metric differenceswithin categorical boundaries. With the appropriate routines for matching shape representations toobject memory (here, Gaussian RFs with varying standard deviations), the resultingmetric/categorical representation supports classification at multiple levels of abstraction as a naturalconsequence. Simulation results show that, counter to popular wisdom, structural descriptions neednot be limited to recognition at the level of object classes.

The MetriCat approach to shape classification also suggests a unified account of the role(and mechanisms) of visual attention in shape perception. Due to the logistic representation ofmetric properties, metric differences within categorical boundaries are reduced in magnitude relativeto metric differences across boundaries. As a result, the former are more sensitive to noise than arethe latter. MetriCat uses attention to recover such purely metric differences and classify objects asindividual instances. In this role, attention serves to increase the signal-to-noise ratio in therepresentation of a geon, allowing small metric differences to express themselves. The samemechanism serves to maintain the dynamic binding of shape attributes and relations into sets. Inboth cases, the mechanism is a simple parameter, α, that determines the strength with which geonsinhibit one another in the competition for the opportunity to fire: The greater the inhibition, thecleaner the asynchrony, and the higher the signal-to-noise ratio.

Given the benefits of attention to MetriCat, it is reasonable to ask why α should even be avariable: Why not leave it set to a maximum value all the time? The reason is that the privilege offiring cleanly out of synchrony is a finite resource (Hummel & Biederman, 1991). To the extentthat one geon is firing all by itself (and thereby maximizing its binding and its signal-to-noise ratio),then by definition, no other geons can be firing at all. All the simulations reported here were runwith stimuli consisting of single objects. In such situations, it is sensible to leave α at a maximumvalue, devoting as much processing as possible to the one object in the visual field. But in morerealistic situations there are many objects in the visual field, so it may be sensible as a default todevote only a moderate amount of attention to any one object. Such a default would permit rapidclassification at the class-level without necessarily excluding the processing of all other objects inthe visual field (cf. Stankiewicz et al., 1997). When it becomes necessary to recognize an object asan instance, it may be worth "turning attention up," temporarily processing that object to theexclusion of others. Clearly, at this point these considerations are merely speculation. ButMetriCat suggests a specific mechanism for thinking about the implementation and consequencesof visual attention in shape perception.

Relation to Other Models of Object RecognitionMetriCat bears important similarities to current view-based models of object recognition.

First, like the models of Poggio and his colleagues (e.g., Edelman et al., 1996; Poggio & Edelman,1990; Vetter et al., 1994), MetriCat uses units with Gaussian receptive fields as the basis for objectclassification. However, it is important to recognize that MetriCat differs markedly from any view-based model in the representational space over which the Gaussians are defined. In all extantview-based models, shape classification is performed by basis functions defined in the space of the2-D coordinates of object features: The representational assumption underlying these models is thatobjects are represented and matched to memory in terms of the coordinates of their features (hence

the name "view-based"). The psychological plausibility of this assumption is questionable(Hummel & Stankiewicz, 1996b). By contrast, in MetriCat, the Gaussian RFs are defined in thespace of categorical attributes and relations. As a consequence, the models have very differentproperties. In particular, it is not clear how view-based models could be adapted to account for therole of categorical properties in shape perception (as summarized in the Introduction; see alsoBiederman & Gerhardstein, 1995). More importantly, the representations in MetriCat arestructured (i.e., attributes are represented independently of their relations) whereas current view-based models are strictly holistic. This difference has important implications for the models' abilityto generalize from known to novel instances (cf. Fodor & Pylyshyn, 1988; Hummel & Biederman,1992; Hummel & Stankiewicz, 1996b).

A second similarity between MetriCat and view-based models is the shared emphasis onmetric properties in the representation of object shape. But here, too, the differences are moreimportant than the similarities. The most important difference is that in view-based models, therepresentational space (feature coordinates) is perfectly linear in (indeed, identical to) the perceptualdomain from which it derives (feature coordinates as they appear in the image). By contrast, inMetriCat, the representational space (the logistic representation attribute values) is nonlinear in theperceptual domain from which it derives (the numerical values of the attributes). This difference isimportant because the nonlinearity plays a critical role in the behavior of MetriCat, and the linearityplays an equally critical role in the view-based models. It is doubtful that the principles underlyingMetriCat could be adapted to operate with a linear representation of attributes, and it is equallydoubtful that the principles underlying the view-based approach could be adapted to work with anonlinear representation of coordinates (Hummel & Stankiewicz, 1996b).

ExtensionsThe simulations reported here suggest that MetriCat has promise as an account of our

ability to recognize objects at multiple levels of abstraction. However, numerous additional tests arerequired to evaluate MetriCat as a general model of object recognition. Doing so will requireintegrating MetriCat with a front-end (such as the early layers of JIM) that can automaticallyrecover geon attributes and relations from line drawings or gray-level images. At this point, it isnonetheless possible to speculate about some other implications of the MetriCat approach to objectrecognition. One particularly interesting implication concerns the relationship between levels ofclassification and sensitivity to variations in viewpoint. In general, categorical attributes andrelations are more robust to variations in viewpoint than otherwise equivalent metric attributes(Biederman, 1987). For example, a representation that describes the axis of a curved cone simplyas "curved" will remain the same in many views of the cone, whereas a representation that specifiesits numerical degree of curvature (e.g., as given by the curvature of the cone's bounding edges) willchange as the cone is rotated in depth. MetriCat relies more heavily on precise metric properties forinstance-level recognition than for class-level recognition. (This is true because of the way it codesnumerical attributes and the way it matches those attributes to memory for recognition.) It thereforepredicts that instance-level recognition will tend to be more view-sensitive than class-levelrecognition. There is some indirect support for this prediction (see Biederman & Gerhardstein,1995; Bülthoff et al., 1995; Tarr & Bülthoff, 1995).

MetriCat as Model of Visual AttentionAs a model of attention, MetriCat is still in an early stage of development. The mechanism

it uses to implement attentional selection—enhanced inhibition for asynchrony—is motivatedstrictly by the problem of generating structural descriptions in a neural network for shaperecognition. Conspicuously absent from the discussion so far is any basis for deciding whenattention should be directed to an object and where to direct it. One particularly interesting propertyof MetriCat is that an answer to the first question (How do we know when to direct attention?) mayserve as a partial answer to the second (How do we know where to direct it?). Note that α is aglobal parameter. It does not operate on any particular location in the visual field, or even on anyparticular object. Rather, it serves only to enhance the ability of whatever is currently firing toinhibit anything else that might attempt to fire in the near future: α is directed in time rather than

space. Any given object (or part) exists at a particular location in the visual field, and fires at aparticular time. Thus, knowing when to direct attention is tantamount to knowing where to direct it:Directing attention to whatever is firing at a given time implicitly directs attention to a particularobject occupying a particular location in the visual field.

How can MetriCat figure out when to direct its attention? Part of the answer to thisquestion must depend on the system's processing goals, and this part of the question is clearlybeyond the scope of our current research. However, given that the system knows that it cares abouta particular object, there is a straightforward basis for dynamically directing attention to variousparts of that object. This basis relates to the relationship between the activation of object units andthe utility of object parts for discriminating one object from another. Ambiguity about an object'sidentity consists of simultaneous activation in multiple, inconsistent object units. For example, ifthe units for both "teapot" and "lamp" are equally active at some instant, then at that instant,MetriCat is undecided about whether the object is a teapot or a lamp. Imagine that the system isprocessing some (as yet unrecognized) stimulus, and that, on the basis of the geon that is currentlyfiring, there is no way to decide whether the object is a teapot or a lamp. As long as the current(uninformative) geon continues to fire, the activations of the "lamp" and "teapot" units will remainunchanged. Then the current geon stops firing, and some new geon starts to fire. If this new geonis informative, then the activations of the object units will begin to change—say, "teapot" begins todecline and "lamp" begins to grow. This change in activation can serve as a signal that the geonunder consideration is informative, and that attention should be directed to it (so it can inhibit itscompetitors and remain active longer without interference). Once the informative geon has done allit can in the way of indicating whether the object is a teapot or a lamp, the object units' activationswill once again stop changing. Attention should now be moved away from this geon and onto adifferent one. Thus, one way to dynamically direct attention may be to set α to a value proportionalto the change in the activation of object units. We have implemented an early version of thismechanism and the results so far have been promising. Augmented with a learning algorithm forkeeping track of which parts are diagnostic for the recognition of which objects, this temporalsearch mechanism might serve as one way to search for diagnostic features for object recognition(e.g., as discussed by Biederman, 1987).

Of course, it is unlikely that this simple mechanism could constitute a complete solution tothe problem of allocating visual attention. Among other things, it provides no clear way to explainphenomena such as visual search or visual neglect, which seem to have a decidedly spatialcomponent. However, it is tempting to speculate that this type of mechanism might constitute atleast one basis for attentional selection in human shape perception.

Selective Processing of Visual "Features"One other extension of MetriCat is worth mentioning here. MetriCat uses Gaussian

receptive fields with variable standard deviations (σ) to classify objects at multiple levels ofabstraction. In the model's current form, σ for any given unit is a scalar. Implicitly, this conventionsets σ to the same value for all attribute dimensions. An extension (and generalization) of thisapproach would be to make σ a vector that can take different values on different dimensions (as p iscurrently). For example, a given GFA might select for cones with a very particular aspect ratio (i.e.,by setting σ to a small value on the dimension aspect ratio), without being as selective for particularvalues on other dimensions (σ would be larger on these dimensions). This kind of selectivenarrowing of the units' receptive fields would allow MetriCat to choose diagnostic stimulusdimensions for instance-level classification, and constitutes another way the architecture mightaccount for the use of diagnostic features for visual scrutiny.

Summary and ConclusionsMetriCat is in an early stage of development, but the approach is promising. The

fundamental principles underlying the model are that: (1) Objects are visually represented ascollections of part attributes and relations, which are dynamically bound into structural descriptions(Hummel & Biederman, 1992); (2) part attributes and relations are represented in a nonlinear

fashion that emphasizes categorical boundaries without discarding all metric information(Stankiewicz & Hummel, 1996); and (3) attention enables parts to inhibit one another so that theycan fire cleanly out of synchrony. Together, these principles provide a preliminary account of thehuman capacity to classify objects at multiple levels of abstraction, and suggest a theory of themechanism of attention in shape perception.

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Appendix

Oscillatory Gates: Each gate, i, has two components, an excitor, Ei, and an inhibitor, Ii,which interact to cause Ei to oscillate. Ei changes according to:

∆Ei = 0.9 1 − Ei( ) 1− 2Ii( )[ ] − 0.51 − Ei( ) − Ljj ≠ i∑ , (5)

where Ii is the activation of the corresponding inhibitor, and Lj is the inhibitory strength of gate j.Ii changes according to:

∆Ii =

0.005 Ei( ) iff Ii < 0.2

0.5 Ei( ) iff Ii ≥ 0.2

if ri

−0.01 iff Ii > 0.9

-0.5 iff Ii ≤ 0.9

, otherwise

. (6)

where ri is a Boolean variable that is set to true whenever Ii < 0.1 and to false whenever Ii > 0.995;ri remains at its last set value whenever 0.1 < Ii < 0.995. Together, (5) and (6) cause Ei to oscillate,as illustrated in Figure 3 (Hummel & Stankiewicz, 1996a; Hummel & Holyoak, 1997). Incombination with the inhibition between competing gates (L), the result is that separate Ei tend tooscillate out of synchrony with one another. Li, the inhibitory strength of gate i is:

Li = EiPi , (7)

where Pi —the priority of gate i—modulates the ability of i to inhibit other gates, j, and α modulatesthe global ability of all gates to inhibit one another. A gate's priority is:

Pi =0 if I

i> 0.2 and ∆I

i=(-0.5),

Pi = Pi + 0.001 + 0.01 1− Pi( )[ ]

otherwise.

(8)

By (6) and (8), Pi will go to zero immediately after i fires and then gradually grow toward 1.0. Pihelps the gates to "time share" by ensuring that a gate will have little ability to inhibit other gates(and therefore little opportunity to fire) if it has fired in the recent past. α corresponds to theamount of attention directed to an object. When α is large (e.g., 1.5), the various gates inhibit oneanother strongly (Eq. 7) and therefore fire cleanly out of synchrony; when α is zero, the gatescannot inhibit one another at all, and therefore never fire out of synchrony; at intermediate values ofα, the gates fire somewhat out of synchrony.

Attribute Units: Each attribute unit, i, has a Gaussian receptive field (RF) with a center, i,in the range of logistic values (0.0..1.0) and a standard deviation, i, in the range 0.0..0.25. Theinput to attribute unit i, from all geons, j, is:

Ii = E jG i − l j , i

j∑ , (9)

where Ej is the value of the excitatory gate on geon j, lj is the logistic value of geon j on thecorresponding attribute, and G() is the Gaussian. The activation of an attribute unit grows as:

∆ai = 0.5 Ii − 0.05ai . (10)

GFA Units: GFA units have Gaussian RFs in the 350-dimensional (50 units X 7attributes) space of attribute units. The activation of GFA unit i in response to attribute vector a is:

Ai = Gpi − a

d

, i

, (11)

where pi is the mean (center) of i's RF, i is its standard deviation, and d is the dimensionality of theattribute vector (here, 350). The output of GFA unit i at time t is:

Oit =

Ait if A

it > O

it −1

Oit −1 otherwise

. (12)

Object Units: Object units have Gaussian RFs in the space of GFA unit outputs. Theactivation of object unit i is:

Bi = Gp

i− o

pi

, i

, (13)

where pi is the mean (center) of i's RF, o is the vector of GFA unit outputs, and i is the standarddeviation of i's receptive field. The normalization by ||pi|| (the length of the object unit's receptivefield vector) serves to correct for the dimensionality of the GFA vector. (Note that, as more objectsare learned, the number of object parts stored in memory grows, changing the dimensionality ofboth the GFA output vector and an object unit's receptive field.)